Degenerate stochastic differential equations for catalytic branching networks
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 943-980.

On prouve l'unicité d'un problème de martingale correspondant à une EDS dégénerée, qui apparaît comme un modèle de réseaux avec branchement catalytique. Ce travail est une extension des résultats de Dawson et Perkins [Illinois J. Math. 50 (2006) 323-383] au cas de réseaux généraux. On obtient en particulier des estimées pour le semi-groupe des réseaux généraux, sous forme de normes de Hölder pondérées; et on établit l'équivalence de ces normes avec des normes de semi-groupe dans ce contexte général.

Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of the paper by Dawson and Perkins [Illinois J. Math. 50 (2006) 323-383] to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Hölder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting.

DOI : 10.1214/08-AIHP186
Classification : 60J60, 60J80, 60J35
Mots-clés : stochastic differential equations, martingale problem, degenerate operators, catalytic branching networks, diffusions, semigroups, weighted Hölder norms, perturbations 
@article{AIHPB_2009__45_4_943_0,
     author = {Kliem, Sandra},
     title = {Degenerate stochastic differential equations for catalytic branching networks},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {943--980},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {4},
     year = {2009},
     doi = {10.1214/08-AIHP186},
     mrnumber = {2572159},
     zbl = {1201.60058},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/08-AIHP186/}
}
TY  - JOUR
AU  - Kliem, Sandra
TI  - Degenerate stochastic differential equations for catalytic branching networks
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 943
EP  - 980
VL  - 45
IS  - 4
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/08-AIHP186/
DO  - 10.1214/08-AIHP186
LA  - en
ID  - AIHPB_2009__45_4_943_0
ER  - 
%0 Journal Article
%A Kliem, Sandra
%T Degenerate stochastic differential equations for catalytic branching networks
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 943-980
%V 45
%N 4
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/08-AIHP186/
%R 10.1214/08-AIHP186
%G en
%F AIHPB_2009__45_4_943_0
Kliem, Sandra. Degenerate stochastic differential equations for catalytic branching networks. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 943-980. doi : 10.1214/08-AIHP186. http://archive.numdam.org/articles/10.1214/08-AIHP186/

[1] S. R. Athreya, M. T. Barlow, R. F. Bass and E. A. Perkins. Degenerate stochastic differential equations and super-Markov chains. Probab. Theory Related Fields 123 (2002) 484-520. | MR | Zbl

[2] S. R. Athreya, R. F. Bass and E. A. Perkins. Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces. Trans. Amer. Math. Soc. 357 (2005) 5001-5029 (electronic). | MR | Zbl

[3] R. F. Bass. Diffusions and Elliptic Operators. Springer, New York, 1998. | MR | Zbl

[4] R. F. Bass and E. A. Perkins. Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains. Trans. Amer. Math. Soc. 355 (2003) 373-405 (electronic). | MR | Zbl

[5] R. F. Bass and E. A. Perkins. Degenerate stochastic differential equations arising from catalytic branching networks. Electron. J. Probab. 13 (2008) 1808-1885. | MR

[6] D. A. Dawson, A. Greven, F. Den Hollander, R. Sun and J. M. Swart. The renormalization transformation for two-type branching models. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 1038-1077. | Numdam | MR | Zbl

[7] D. A. Dawson and E. A. Perkins. On the uniqueness problem for catalytic branching networks and other singular diffusions. Illinois J. Math. 50 (2006) 323-383 (electronic). | MR | Zbl

[8] M. Eigen and P. Schuster. The Hypercycle: A Principle of Natural Self-organization. Springer, Berlin, 1979.

[9] J. Hofbauer and K. Sigmund. The Theory of Evolution and Dynamical Systems. London Math. Soc. Stud. Texts 7. Cambridge Univ. Press, Cambridge, 1988. | MR | Zbl

[10] L. Mytnik. Uniqueness for a mutually catalytic branching model. Probab. Theory Related Fields 112 (1998) 245-253. | MR | Zbl

[11] E. A. Perkins. Dawson-Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999) 125-324. Lecture Notes in Math. 1781. Springer, Berlin, 2002. | MR | Zbl

[12] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Vol. 2. Reprint of the 2nd (1994) edition. Cambridge Univ. Press, Cambridge, 2000. | MR | Zbl

[13] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Grundlehren Math. Wiss. 233. Springer, Berlin, 1979. | MR | Zbl

Cité par Sources :